Question
The law of cosines reduces to the Pythagorean Theorem whenever the triangle is acute.
True or false
True or false
Answers
GPT-4o mini
False. The law of cosines reduces to the Pythagorean theorem specifically when the triangle is a right triangle, not just when it is acute. The Pythagorean theorem applies only in the case where one of the angles is exactly \(90^\circ\).
In general, the law of cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
If angle \(C\) is \(90^\circ\), then \(\cos(90^\circ) = 0\), and the equation simplifies to:
\[
c^2 = a^2 + b^2
\]
which is the Pythagorean theorem.
In an acute triangle, the cosine of each angle is positive, but that does not mean the law of cosines simplifies to the Pythagorean theorem.
In general, the law of cosines states:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
If angle \(C\) is \(90^\circ\), then \(\cos(90^\circ) = 0\), and the equation simplifies to:
\[
c^2 = a^2 + b^2
\]
which is the Pythagorean theorem.
In an acute triangle, the cosine of each angle is positive, but that does not mean the law of cosines simplifies to the Pythagorean theorem.